JSH: Prime gaps examples
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From: Joshua Cranmer on 22 Jul 2010 17:40 On 07/22/2010 10:28 AM, JSH wrote: > On Jul 21, 9:37 pm, Tim Little<t...(a)little-possums.net> wrote: >> When you count the primes less than K with residue 1 vs residue 2 >> mod 3, the 2s are in front for almost all values of K. Why is that >> if your claim of "no preference" was correct? > > Why can you have runs of 10 heads in a row when you flip a coin? Because that is a possible outcome of flipping a coin several times. > What does that prove mathematically? Coin bias towards heads? Actually, statistically speaking, flipping coins will produce some surprisingly long runs of all heads or all tails. Most people, if inventing a pseudorandom sequence themselves, don't realize this and have too few long runs. There are, however, statistical tests that indicate how well a given model explains an observed occurrence. If you flip a coin ten times in a row and get ten heads, it is not likely to be fair, but if you flip that coin a million times in a row and get a run of ten heads, it can certainly still be fair. >> What *in mathematical language* are you actually claiming? > > With no preference the primes behave probabilistically with regard to > residues modulo a lesser prime. That is the prime residue axiom. That statement is completely vacuous. You need to specify your probability distribution (any nonnegative function whose integral over its domain is exactly 1). -- Beware of bugs in the above code; I have only proved it correct, not tried it. -- Donald E. Knuth
From: Tim Little on 23 Jul 2010 21:56 On 2010-07-22, JSH <jstevh(a)gmail.com> wrote: > On Jul 21, 9:37 pm, Tim Little <t...(a)little-possums.net> wrote: >> When you count the primes less than K with residue 1 vs residue 2 >> mod 3, the 2s are in front for almost all values of K. Why is that >> if your claim of "no preference" was correct? > > Why can you have runs of 10 heads in a row when you flip a coin? Who said anything about a finite number of "heads" like 10, apart form you? This is a well-known result that holds out to infinity. If there were no bias in a random sequence, the probability of such a result would be zero. Not just "smallish" (like a run of 10 heads), but zero. >> What *in mathematical language* are you actually claiming? > > With no preference the primes behave probabilistically with regard > to residues modulo a lesser prime. That is the prime residue axiom. That's still not a mathematical statement. What sort of mathematical term is "behave"? Be specific: exactly what properties are you claiming to be probabilistic, and exactly what laws of probability do you claim they satisfy? - Tim
From: Tim Little on 23 Jul 2010 21:59 On 2010-07-22, Jesse F. Hughes <jesse(a)phiwumbda.org> wrote: > I wonder if the phrase "for almost all values of K" means anything. More technically, it means that the limit density of counterexamples is 0. If the most obvious interpretation of James' ambiguous conjecture (that he calls an axiom) were true, the limit density would be 1/2. - Tim
From: JSH on 23 Jul 2010 23:22 On Jul 23, 6:56 pm, Tim Little <t...(a)little-possums.net> wrote: > On 2010-07-22, JSH <jst...(a)gmail.com> wrote: > > > On Jul 21, 9:37 pm, Tim Little <t...(a)little-possums.net> wrote: > >> When you count the primes less than K with residue 1 vs residue 2 > >> mod 3, the 2s are in front for almost all values of K. Why is that > >> if your claim of "no preference" was correct? > > > Why can you have runs of 10 heads in a row when you flip a coin? > > Who said anything about a finite number of "heads" like 10, apart form > you? > > This is a well-known result that holds out to infinity. If there were Interesting. There may be a simple reason. What if you drop the first 10 primes? Does it hold then for the primes thereafter? > no bias in a random sequence, the probability of such a result would > be zero. Not just "smallish" (like a run of 10 heads), but zero. Well yeah, seems reasonable, if true, but I've seen enough nutty stuff that math people claim is proven that actually isn't that my expectations are not high. James Harris
From: Tim Little on 24 Jul 2010 03:21
On 2010-07-24, JSH <jstevh(a)gmail.com> wrote: > What if you drop the first 10 primes? Does it hold then for the > primes thereafter? It holds if you drop any finite number of primes. > Well yeah, seems reasonable, if true, but I've seen enough nutty > stuff that math people claim is proven that actually isn't that my > expectations are not high. I haven't seen any, but I don't count you as a math person. If I did, I would have seen lots. - Tim |